On bases, finite dimensional decompositions and weaker structures in Banach spaces

This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., that X has a basis if X* does, and that if X has a basis, then X* has a basis provided that X* is separable and satisfies Grothendieck's approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separable ℒ _p space has a basis.